Physics and Philosophy

2The History of Quantum Theory

THE origin of quantum theory is connected with a well-known phenomenon,
which did not belong to the central parts of atomic physics. Any
piece of matter when it is heated starts to glow, gets red hot
and white hot at higher temperatures. The colour does not depend
much on the surface of the material, and for a black body it depends
solely on the temperature. Therefore, the radiation emitted by
such a black body at high temperatures is a suitable object for
physical research; it is a simple phenomenon that should find
a simple explanation in terms of the known laws for radiation
and heat. The attempt made at the end of the nineteenth century
by Lord Rayleigh and Jeans failed, however, and revealed serious
difficulties. It would not be possible to describe these difficulties
here in simple terms. It must be sufficient to state that the
application of the known laws did not lead to sensible results.
When Planck, in 1895, entered this line of research he tried to
turn the problem from radiation to the radiating atom. This turning
did not remove any of the difficulties inherent in the problem,
but it simplified the interpretation of the empirical facts. It
was just at this time, during the summer of 1900, that Curlbaum
and Rubens in Berlin had made very accurate new measurements of
the spectrum of heat radiation. When Planck heard of these results
he tried to represent them by simple mathematical formulas which
looked plausible from his research on the general connection between
heat and radiation. One day Planck and Rubens met for tea in Planck's
home and compared Rubens' latest results with a new formula suggested
by Planck. The comparison showed a complete agreement. This was
the discovery of Planck's law of heat radiation.

It was at the same time the beginning of intense theoretical work
for Planck. What was the correct physical interpretation of the
new formula? Since Planck could, from his earlier work, translate
his formula easily into a statement about the radiating atom (the
so-called oscillator), he must soon have found that his formula
looked as if the oscillator could only contain discrete quanta
of energy - a result that was so different from anything known
in classical physics that he certainly must have refused to believe
it in the beginning. But in a period of most intensive work during
the summer of 1900 he finally convinced himself that there was
no way of escaping from this conclusion. It was told by Planck's
son that his father spoke to him about his new ideas on a long
walk through the Grunewald, the wood in the suburbs of Berlin.
On this walk he explained that he felt he had possibly made a
discovery of the first rank, comparable perhaps only to the discoveries
of Newton. So Planck must have realised at this time that his
formula had touched the foundations of our description of nature,
and that these foundations would one day start to move from their
traditional present location toward a new and as yet unknown position
of stability. Planck, who was conservative in his whole outlook,
did not like this consequence at all, but he published his quantum
hypothesis in December of 1900.

The idea that energy could be emitted or absorbed only in discrete
energy quanta was so new that it could not be fitted into the
traditional framework of physics. An attempt by Planck to reconcile
his new hypothesis with the older laws of radiation failed in
the essential points. It took five years until the next step could
be made in the new direction.

This time it was the young Albert Einstein, a revolutionary genius
among the physicists, who was not afraid to go further away from
the old concepts. There were two problems in which he could make
use of the new ideas. One was the so-called photoelectric effect,
the emission of electrons from metals under the influence of light.
The experiments, especially those of Lenard, had shown that the
energy of the emitted electrons did not depend on the intensity
of the light, but only on its colour or, more precisely, on its
frequency. This could not be understood on the basis of the traditional
theory of radiation. Einstein could explain the observations by
interpreting Planck's hypothesis as saying that light consists
of quanta of energy travelling through space. The energy of one
light quantum should, in agreement with Planck's assumptions,
be equal to the frequency of the light multiplied by Planck's
constant.

The other problem was the specific heat of solid bodies. The traditional
theory led to values for the specific heat which fitted the observations
at higher temperatures but disagreed with them at low ones. Again
Einstein was able to show that one could understand this behaviour
by applying the quantum hypothesis to the elastic vibrations of
the atoms in the solid body. These two results marked a very important
advance, since they revealed the presence of Planck's quantum
of action - as his constant is called among the physicists - in
several phenomena, which had nothing immediately to do with heat
radiation. They revealed at the same time the deeply revolutionary
character of the new hypothesis, since the first of them led to
a description of light completely different from the traditional
wave picture. Light could either be interpreted as consisting
of electromagnetic waves, according to Maxwell's theory, or as
consisting of light quanta, energy packets travelling through
space with high velocity. But could it be both? Einstein knew,
of course, that the well-known phenomena of diffraction and interference
can be explained only on the basis of the wave picture. He was
not able to dispute the complete contradiction between this wave
picture and the idea of the light quanta; nor did he even attempt
to remove the inconsistency of this interpretation. He simply
took the contradiction as something which would probably be understood
only much later.

In the meantime the experiments of Becquerel, Curie and Rutherford
had led to some clarification concerning the structure of the
atom. In 1911 Rutherford's observations on the interaction of
a-rays penetrating through matter resulted in his famous atomic
model. The atom is pictured as consisting of a nucleus, which
is positively charged and contains nearly the total mass of the
atom, and electrons, which circle around the nucleus like the
planets circle around the sun. The chemical bond between atoms
of different elements is explained as an interaction between the
outer electrons of the neighbouring atoms; it has not directly
to do with the atomic nucleus. The nucleus determines the chemical
behaviour of the atom through its charge which in turn fixes the
number of electrons in the neutral atom. Initially this model
of the atom could not explain the most characteristic feature
of the atom, its enormous stability. No planetary system following
the laws of Newton's mechanics would ever go back to its original
configuration after a collision with another such system. But
an atom of the element carbon, for instance, will still remain
a carbon atom after any collision or interaction in chemical binding.

The explanation for this unusual stability was given by Bohr in
1913, through the application of Planck's quantum hypothesis.
If the atom can change its energy only by discrete energy quanta,
this must mean that the atom can exist only in discrete stationary
states, the lowest of which is the normal state of the atom. Therefore,
after any kind of interaction the atom will finally always fall
back into its normal state.

By this application of quantum theory to the atomic model, Bohr
could not only explain the stability of the atom but also. in
some simple cases, give a theoretical interpretation of the line
spectra emitted by the atoms after the excitation through electric
discharge or heat. His theory rested upon a combination of classical
mechanics for the motion of the electrons with quantum conditions,
which were imposed upon the classical motions for defining the
discrete stationary states of the system. A consistent mathematical
formulation for those conditions was later given by Sommerfeld.
Bohr was well aware of the fact that the quantum conditions spoil
in some way the consistency of Newtonian mechanics. In the simple
case of the hydrogen atom one could calculate from Bohr's theory
the frequencies of the light emitted by the atom, and the agreement
with the observations was perfect. Yet these frequencies were
different from the orbital frequencies and their harmonies of
the electrons circling around the nucleus, and this fact showed
at once that the theory was still full of contradictions. But
it contained an essential part of the truth. It did explain qualitatively
the chemical behaviour of the atoms and their line spectra; the
existence of the discrete stationary states was verified by the
experiments of Franck and Hertz, Stern and Gerlach.

Bohr's theory had opened up a new line of research. The great
amount of experimental material collected by spectroscopy through
several decades was now available for information about the strange
quantum laws governing the motions of the electrons in the atom.
The many experiments of chemistry could be used for the same purpose.
It was from this time on that the physicists learned to ask the
right questions; and asking the right question is frequently more
than halfway to the solution of the problem.

What were these questions? Practically all of them had to do with
the strange apparent contradictions between the results of different
experiments. How could it be that the same radiation that produces
interference patterns, and therefore must consist of waves, also
produces the photoelectric effect, and therefore must consist
of moving particles? How could it be that the frequency of the
orbital motion of the electron in the atom does not show up in
the frequency of the emitted radiation? Does this mean that there
is no orbital motion? But if the idea of orbital motion should
be incorrect, what happens to the electrons inside the atom? One
can see the electrons move through a cloud chamber, and sometimes
they are knocked out of an atom- why should they not also move
within the atom? It is true that they might be at rest in the
normal state of the atom, the state of lowest energy. But there
are many states of higher energy, where the electronic shell has
an angular momentum. There the electrons cannot possibly be at
rest. One could add a number of similar examples. Again and again
one found that the attempt to describe atomic events in the traditional
terms of physics led to contradictions.

Gradually, during the early twenties, the physicists became accustomed
to these difficulties, they acquired a certain vague knowledge
about where trouble would occur, and they learned to avoid contradictions.
They knew which description of an atomic event would be the correct
one for the special experiment under discussion. This was not
sufficient to form a consistent general picture of what happens
in a quantum process, but it changed the minds of the physicists
in such a way that they somehow got into the spirit of quantum
theory. Therefore, even some time before one had a consistent
formulation of quantum theory one knew more or less what would
be the result of any experiment.

One frequently discussed what one called ideal experiments. Such
experiments were designed to answer a very critical question irrespective
of whether or not they could actually be carried out. Of course
it was important that it should be possible in principle to carry
out the experiment, but the technique might be extremely complicated.
These ideal experiments could be very useful in clarifying certain
problems. If there was no agreement among the physicists about
the result of such an ideal experiment, it was frequently possible
to find a similar but simpler experiment that could be carried
out, so that the experimental answer contributed essentially to
the clarification of quantum theory.

The strangest experience of those years was that the paradoxes
of quantum theory did not disappear during this process of clarification;
on the contrary, they became even more marked and more exciting.
There was, for instance, the experiment of Compton on the scattering
of X-rays. From earlier experiments on the interference of scattered
light there could be no doubt that scattering takes place essentially
in the following way: The incident light wave makes an electron
in the beam vibrate in the frequency of the wave; the oscillating
electron then emits a spherical wave with the same frequency and
thereby produces the scattered light. However, Compton found in
1923 that the frequency of scattered X-rays was different from
the frequency of the incident X-ray. This change of frequency
could be formally understood by assuming that scattering is to
be described as collision of a light quantum with an electron.
The energy of the light quantum is changed during the collision;
and since the frequency times Planck's constant should be the
energy of the light quantum, the frequency also should be changed.
But what happens in this interpretation of the light wave? The
two experiments - one on the interference of scattered light and
the other on the change of frequency of the scattered light -
seemed to contradict each other without any possibility of compromise.

By this time many physicists were convinced that these apparent
contradictions belonged to the intrinsic structure of atomic physics.
Therefore, in I924 de Broglie in France tried to extend the dualism
between wave description and particle description to the elementary
particles of matter, primarily to the electrons. He showed that
a certain matter wave could 'correspond' to a moving electron,
just as a light wave corresponds: to a moving light quantum. It
was not clear at the time what the word 'correspond' meant in
this connection. But de Broglie suggested that the quantum condition
in Bohr's theory should be interpreted as a statement about the
matter waves. A wave circling around a nucleus can for geometrical
reasons only be a stationary wave; and the perimeter of the orbit
must be an integer multiple of the wave length. In this way de
Broglie's idea connected the quantum condition. which always had
been a foreign element in the mechanics of the electrons, with
the dualism between waves and particles.

In Bohr's theory the discrepancy between the calculated orbital
frequency of the electrons and the frequency of the emitted radiation
had to be interpreted as a limitation to the concept of the electronic
orbit. This concept had been somewhat doubtful from the beginning.
For the higher orbits, however, the electrons should move at a
large distance from the nucleus just as they do when one sees
them moving through a cloud chamber. There one should speak about
electronic orbits. It was therefore very satisfactory that for
these higher orbits the frequencies of the emitted radiation approach
the orbital frequency and its higher harmonics. Also Bohr had
already suggested in his early papers that the intensities of
the emitted spectral lines approach the intensities of the corresponding
harmonics. This principle of correspondence had proved very useful
for the approximative calculation of the intensities of spectral
lines. In this way one had the impression that Bohr's theory gave
a qualitative but not a quantitative description of what happens
inside the atom; that some new feature of the behaviour of matter
was qualitatively expressed by the quantum conditions, which in
turn were connected with the dualism between waves and particles.

The precise mathematical formulation of quantum theory finally
emerged from two different developments. The one started from
Bohr's principle of correspondence. One had to give up the concept
of the electronic orbit, but still had to maintain it in the limit
of high quantum numbers, i.e., for the large orbits.

In this latter case the emitted radiation, by means of its frequencies
and intensities, gives a picture of the electronic orbit; it represents
what the mathematicians call a Fourier expansion of the orbit.
The idea suggested itself that one should write down the mechanical
laws not as equations for the positions and velocities of the
electrons but as equations for the frequencies and amplitudes
of their Fourier expansion. Starting from such equations and changing
them very little one could hope to come to relations for those
quantities which correspond to the frequencies and intensities
of the emitted radiation, even for the small orbits and the ground
state of the atom. This plan could actually be carried out; in
the summer of 1925 it led to a mathematical formalism called matrix
mechanics or, more generally, quantum mechanics. The equations
of motion in Newtonian mechanics were replaced by similar equations
between matrices; it was a strange experience to find that many
of the old results of Newtonian mechanics, like conservation of
energy, etc., could be derived also in the new scheme. Later the
investigations of Born, Jordan and Dirac showed that the matrices
representing position and momentum of the electron did not commute.
This latter fact demonstrated clearly the essential difference
between quantum mechanics and classical mechanics.

The other development followed de Broglie's idea of matter waves.
Schrödinger tried to set up a wave equation for de Broglie's
stationary waves around the nucleus. Early in 1926 he succeeded
in deriving the energy values of the stationary states of the
hydrogen atom as 'Eigenvalues' of his wave equation and could
give a more general prescription for transforming a given set
of classical equations of motion into a corresponding wave equation
in a space of many dimensions. Later he was able to prove that
his formalism of wave mechanics was mathematically equivalent
to the earlier formalism of quantum mechanics.

Thus one finally had a consistent mathematical formalism, which
could be defined in two equivalent ways starting either from relations
between matrices or from wave equations. This formalism gave the
correct energy values for the hydrogen atom: it took less than
one year to show that it was also successful for the helium atom
and the more complicated problems of the heavier atoms. But in
what sense did the new formalism describe the atom? The paradoxes
of the dualism between wave picture and particle picture were
not solved; they were hidden somehow in the mathematical scheme.

A first and very interesting step toward a real understanding
Of quantum theory was taken by Bohr, Kramers and Slater in 1924.
These authors tried to solve the apparent contradiction between
the wave picture and the particle picture by the concept of the
probability wave. The electromagnetic waves were interpreted not
as 'real' waves but as probability waves, the intensity of which
determines in every point the probability for the absorption (or
induced emission) of a light quantum by an atom at this point.
This idea led to the conclusion that the laws of conservation
of energy and momentum need not be true for the single event,
that they are only statistical laws and are true only in the statistical
average. This conclusion was not correct, however, and the connections
between the wave aspect and the particle aspect of radiation were
still more complicated.

But the paper of Bohr, Kramers and Slater revealed one essential
feature of the correct interpretation of quantum theory. This
concept of the probability wave was something entirely new in
theoretical physics since Newton. Probability in mathematics or
in statistical mechanics means a statement about our degree of
knowledge of the actual situation. In throwing dice we do not
know the fine details of the motion of our hands which determine
the fall of the dice and therefore we say that the probability
for throwing a special number is just one in six. The probability
wave of Bohr, Kramers, Slater, however, meant more than that;
it meant a tendency for something. It was a quantitative version
of the old concept of 'potentia' in Aristotelian philosophy. It
introduced something standing in the middle between the idea of
an event and the actual event, a~~ strange kind of physical reality
just in the middle between possibility and reality. r Later when
the mathematical framework of quantum theory was fixed, Born took
up this idea of the probability wave and gave a clear definition
of the mathematical quantity in the formalism. which was to be
interpreted as the probability wave. It X as not a three-dimensional
wave like elastic or radio waves, but a wave in the many-dimensional
configuration space, and therefore a rather abstract mathematical
quantity.

Even at this time, in the summer of I926, it was not clear in
every case how the mathematical formalism should be used to describe
a given experimental situation. One knew how to describe the stationary
states of an atom, but one did not know how to describe a much
simpler event - as for instance an electron moving through a cloud
chamber.

When Schrödinger in that summer had shown that his formalism
of wave mechanics was mathematically equivalent to quantum mechanics
he tried for some time to abandon the idea of quanta and 'quantum
jumps' altogether and to replace the electrons in the atoms simply
by his three-dimensional matter waves. He was inspired to this
attempt by his result, that the energy levels of the hydrogen
atom in his theory seemed to be simply the eigenfrequencies of
the stationary matter waves. Therefore, he thought it was a mistake
to call them energies: they were just frequencies. But in the
discussions which took place in the autumn of I926 in Copenhagen
between Bohr and Schrödinger and the Copenhagen group of
physicists it soon became apparent that such an interpretation
would not even be sufficient to explain Planck's formula of heat
radiation.

During the months following these discussions an intensive study
of all questions concerning the interpretation of quantum theory
in Copenhagen finally led to a complete and, as many physicists
believe, satisfactory clarification of the situation. But it was
not a solution which one could easily accept. I remember discussions
with Bohr which went through many hours till very late at night
and ended almost in despair; and when at the end of the discussion
I went alone for a walk in the neighbouring park I repeated to
myself again and again the question: Can nature possibly be as
absurd as it seemed to us in these atomic experiments?

The final solution was approached in two different ways. The one
was a turning around of the question. Instead of asking: How can
one in the known mathematical scheme express a given experimental
situation? the other question was put: Is it true, perhaps, that
only such experimental situations can arise in nature as can be
expressed in the mathematical formalism? The assumption that this
was actually true led to limitations in the use of those concepts
that had been the basis of classical physics since Newton. One
could speak of the position and of the velocity of an electron
as in Newtonian mechanics and one could observe and measure these
quantities. But one could not fix both quantities simultaneously
with an arbitrarily high accuracy. Actually the product of these
two inaccuracies turned out to be not less than Planck's constant
divided by the mass of the particle. Similar relations could be
formulated for other experimental situations. They are usually
called relations of uncertainty or principle of indeterminacy.
One had learned that the old concepts fit nature only inaccurately.

lie other way of approach was Bohr's concept of complementarity.
Schrödinger had described the atom as a system not of a nucleus
and electrons but of a nucleus and matter waves. This picture
of the matter waves certainly also contained an element of truth.
Bohr considered the two pictures - particle picture and wave picture
- as two complementary descriptions of the same reality. Any of
these descriptions can be only partially true, there must be limitations
to the use of the particle concept as well as of wave concept,
else one could not avoid contradictions. If one takes into account
those limitations which can be expressed by the uncertainty relations,
the contradictions disappear.

In this way since the spring of I927 one has had a consistent
interpretation of quantum theory, which is frequently called the
'Copenhagen interpretation'. This interpretation received its
crucial test in the autumn of 1927 at the Solvay conference in
Brussels. Those experiments which had always led to the worst
paradoxes were again and again discussed in all details, especially
by Einstein. New ideal experiments were invented to trace any
possible inconsistency of the theory, but the theory was shown
to be consistent and seemed to fit the experiments as far as one
could see.

The details of this Copenhagen interpretation will be the subject
of the next chapter. It should be emphasised at this point that
it has taken more than a quarter of a century to get from the
first idea of the existence of energy quanta to a real understanding
of the quantum theoretical laws. This indicates the great change
that had to take place in the fundamental concepts concerning
reality before one could understand the new situation.